They come from ignorance of the exact starting point, as the dynamics never rolls dice, and the probabilities are the sizes of the sets of microstates a preparation can't tell apart, measured with a ruler nobody was free to choose.
A deterministic world seems to leave no room for quantum probability. The resolution is old and respectable, probability as ignorance of the initial conditions, but that resolution always carried a debt, as somebody has to choose the measure of ignorance, the rule that says how much weight each possible starting point gets, and if you choose it freely you can get any answer you like.
CSD's account starts from the fact that every run of an experiment begins from a slightly different microstate, a slightly different point of the underlying state space, because no procedure controls the arena completely. The set of microstates compatible with what the procedure does control is the preparation region, and the fraction of that region that leads to a given outcome, computed in the one measure the symmetry of the space allows rather than one chosen for convenience, is the outcome's probability. The strong law of large numbers, the theorem that averages over many independent repetitions settle down to the underlying fractions, converts those fractions into the frequencies laboratories actually record.
So there's one path through the world, dice nowhere, and the randomness lies entirely in which microstate each preparation happened to land on. Everyday randomness is a third thing again, and it's worth keeping apart from the quantum kind. A tossed coin, a stirred cup of tea and next week's weather all run on ordinary deterministic physics, and they look random because nobody tracks the fine detail, so tiny differences in how they start grow fast enough that only the broad features stay predictable. CSD has a theorem of that shape for its own state space, which says a broad description of a deterministic path behaves like a chain of random steps once the untracked detail stops carrying information forward, with an error that grows with how many steps you ask about.
That leaves three different things called random and not a dice roll among them, as the path itself is fixed, the quantum probabilities come from not knowing which microstate a preparation landed on, and the everyday kind comes from not following the detail.
This is typicality under repeated preparation, and it's worth stating what it isn't. Probabilities don't come from watching one trajectory for ever and hoping it explores the space fairly, visiting each region in proportion to its size. That property, called ergodicity, is one the corpus proves the projected flow doesn't have, so the time-average route is closed at that level, and the programme never needs it, as each trial is a fresh draw from the conditioned measure and the frequency theorems run on that alone.
The measure-choice debt is paid by a uniqueness theorem. The Fubini-Study measure, the natural volume on the space of quantum states, is the only one invariant under the symmetries of that space, so the ignorance is priced by a ruler nobody selected. The programme separates three regimes of randomness and the dynamics stays deterministic in all three. The path itself has none, the quantum probabilities are the typicality account above, and the apparent randomness of large-scale variables comes from reading a deterministic path through a broad description that throws most of the detail away. The third regime became a theorem in October 2026, which gives the broad description's path probabilities as a chain of one-step probabilities up to an error set by how far the untracked parts have stopped influencing where the microstate is, and that decoupling is a hypothesis the programme supplies nowhere, so the theorem says what follows from decorrelation rather than that the dynamics decorrelates.
Chaos doesn't deliver the quantum probabilities, and the programme checked that rather than assuming it. The route through chaotic waiting times was examined and dropped as the wrong shape for the job, since results of that kind describe sets that are vanishingly rare while the outcome regions of a measurement share out all of the weight between them. The flows the programme uses for isolated systems can't decorrelate at all, as they translate a compact space and so keep returning close to where they began, which is a fact about those particular flows and not about the state space, because the same fibre carries a map whose correlations vanish outright. Whether a realistic de-isolation flow is chaotic stays open and the programme claims nothing either way.
LF1_main_theorem_ae converts region volumes into almost-sure limiting frequencies under i.i.d. preparation sampling, fsMeasure_unique removes the measure freedom, and the general-N volume identities make the limiting values the Born weights.
The non-ergodicity of the epistemic unitary flow is proved and kept as a boundary marker between the claimed route and the unclaimed one. The third regime is abs_measure_historyEvent_sub_markov_le, where the path law of the coarse process factorises into its one-step transitions up to the step count times the decoupling parameter of MicroDecoupled, and cex_not_microDecoupled exhibits a four-point rotation whose coarse process is not Markov for any parameter below one half, so the hypothesis is load-bearing.
The no-mixing boundary is not_hasCorrelationDecay_of_compactGroup, instantiated at the base by not_hasCorrelationDecay_blockPop_of_unitary and at the fibre by not_hasCorrelationDecay_kFlow, against torusDouble_hasCorrelationDecay for the doubling map on that same fibre.
Source links are pinned to a commit, so they do not drift. The anchors above are checked mechanically against the Lean tree on every build. The mathematics is not, and cannot be: that is a human responsibility and it rests with the author.
Part of Constraint-Surface Dynamics · Formalised in csd-lean4.
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